186 On some Extensions of Quaternions, 



(65), stated in its most general form. The general dependence 

 of (65) on (67), or of (56) on (57), is therefore proved to exist; 

 and -the y(»^— w) associative conditions, for which A; = in (54), 

 are seen to be consequences of the J(w'*— w^) other conditions 

 for which k^O; or even of those conditions diminished in num- 

 ber by ^(n*— w), according to what was stated by anticipation 

 in [9.] , and has been proved by the analysis of [14.] . This 

 result is the more satisfactory, because otherwise the conditions 

 of association would essentially involve a system of homogeneous 

 equations of the thi7'd dimension relatively to the symbols {fyh), 

 obtained by substituting in (56) the expressions (95) or (97) for 

 the symbols of the form (J^), including the values (98) of the 

 symbols (/). But we see now (as above stated) that the total 

 number of distinct conditions may be reduced to ^{n'^ — n^) 

 — ^{n^—n), between the total number ^{n^ + n) of constants of 

 multiphcation ; or finally, after the elimination of the ^(n^ + n) 

 symbols of the forms (/) and {fff), to a system of homogeneous 

 equations of the second dimension, namely those determined in 

 [14.], of which the number amounts (as in that paragraph) to 



^{n^-n^)^n''=^n%n + l){n-2), . . (109) 



between the symbols of the form {fyh), whereof the number is 



i{n^ + n)-i{n^ + n)=in%n-l). . . (110) 



[16.] For example, when n=2, the two constants (121) and 

 (122) have been seen in [11.] to be unrestricted by any con- 

 dition. When w=3, we have 9 constants, lately denoted by 

 /j 4 4 ^1 wig m^ n^ 7^2 Wg, wherewith to satisfy 18 homogeneous 

 equations of the second dimension, namely those marked (85) 

 and (86) in [12.] ; which it has been seen to be possible to 

 do, in two distinct ways (A) and (B), and even so as to leave 

 some of the constants arbitrary, in each of the two resulting 

 systems, of associative quadrinomes and tetrads, A similar result 

 has been found by me to hold good for the case 7i = 4, or for 

 the case of associative quines, such as 



P = M; + w: + iey + X^ + /AW, .... (Ill) 



involving four vector-units lkX/j., which obey the laws of con- 

 jugation (32), and of association (51). For although there are 

 in this case only 24= Jw^(n— 1) constants of the form (fyh), to 

 satisfy 80 = J7i^(/i -|- 1) (n — 2) homogeneous equations of the second 

 dimension, yet I have found that the forms* of these equations 



* The subject may be illustrated by the very simple remark, that although 

 the four equations tx=:0, ty=iO, Ma?=0, My=0, are such that no three of 

 them include th^ fourth, since we might (for example) satisfy the three first 

 alone by supposing ^=0, x=0, yet they can all four be satisfied together by 



